CCC '03 S5 - Trucking Troubles

Canadian Computing Competition: 2003 Stage 1, Senior #5

You are a salesperson selling trucks which can carry trucks which can
carry trucks. Needless to say, your trucks are heavy. Also needless to
say, you have to drive one of these trucks across a wide wet domain, and
since it is wet, you need to drive over some bridges. In fact, on every
road between two cities, there is a bridge but there is not a direct
road between every pair of cities. Each bridge can support a certain
maximum weight. This maximum weight is an integer from ~0~ to ~100\,000~.

You have been given a list of cities where there are customers who are
eager to view one of your trucks. These cities are called destination
cities. Since you must decide which truck you will drive through these
cities, you will have to answer the following problem: what is the
maximum weight that can be driven through these destination cities? You
are to write a program to solve this problem.

The first line of input will contain three positive integers: ~c~, ~r~
and ~d~ specifying the number of cities (in total), number of roads
between cities and number of destination cities, respectively. The
cities are numbered from ~1~ to ~c~. There are at most ~10\,000~ cities and at
most ~100\,000~ roads. The next ~r~ lines contain triples ~x\ y\ w~ indicating
that this road runs between city ~x~ and city ~y~ and it has a maximum
weight capacity of ~w~. The next ~d~ lines give the destination cities
you must visit with your truck. There will be at least one destination
city.

You can assume that you are starting in city ~1~ and that city ~1~ is not a
destination city. You can visit the ~d~ destination cities in any order,
but you must visit all ~d~ destination cities.

The output from your program is a single integer, the largest weight
that can be driven through all ~d~ destination cities.